Effects of nanoscale film thickness on apparent
stiffness of and cell-mediated strains in polymers
by
Binu K Oommen
Submitted to the Department of Materials Science and Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Materials Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2006
@ Massachusetts Institute of Technology 2006. All rights reserved.
ARCHIVES
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A uthor .....
....... .......................................
Department of Materials Science and Engineering
April 24, 2006
Certified by
Krystyn J. Van Vliet
Lord
u•undation Assistant Professor
Thesis Supervisor
K?
Accepted by...
S ...........
.-.......
...
..
.Sam uel .
A len..........
Samuel M. Allen
POSCO Professor of Physical Metallurgy
Chair, Departmental Committee on Graduate Students
MASSACHUSETTS INS"TrI
OF TECHNOLOGY
OCT 0 2 2006
LIBRARIES
Effects of nanoscale film thickness on apparent stiffness of
and cell-mediated strains in polymers
by
Binu K Oommen
Submitted to the Department of Materials Science and Engineering
on April 24, 2006, in partial fulfillment of the
requirements for the degree of
Master of Science in Materials Science and Engineering
Abstract
The mechanical properties of compliant materials such as polymeric films and biological membranes that are of nanoscale in thickness are increasingly extracted from
scanning probe microscope-enabled nanoindentation. These films are applied in various fields that require multiaxial loading conditions. The Hertzian contact models
developed for linear elastic materials of semi-infinite thickness fail to accurately predict the elastic modulus E for these compliant materials. This makes it necessary
to understand the evolution of stress and strain fields of these nanoscale structures.
In this thesis we employ computational simulations that are based on experimental
parameters for contact based analysis of compliant polymer thin films, to decouple
the effect of thickness and angle of indentation on calculated mechanical properties.
Traction applied by living cells to these compliant films are studied in detail. We thus
identify the range of strains and material thickness for which contact models could be
used to accurately predict the elastic stiffness of these polymeric films of nanoscale (<
100 nm) thickness using scanning probe microscope-enabled experiments, and the volumes over which adhered cells deform these films. The key results of this thesis enable
accurate experimental analysis of polymeric thin film elastic properties, and design
of synthetic polymeric substrata that will dominate the mechanical environment of
adhered cells.
Thesis Supervisor: Krystyn J. Van Vliet
Title: Lord Foundation Assistant Professor
Acknowledgments
This thesis is a result of several semesters of work whereby I have been accompanied
and supported by many people whom I would like to thank and express my sincere
gratitude.
First of all I would like to thank my advisor Prof. Krystyn van Vliet, whose consistent support, care and enthusiasm towards my research, bears fruit in the form of
this thesis. She devoted many hours of her quality time in bringing out the best out
of me and has all along made a deep impression on me. Besides being an excellent
advisor, she is as close as a relative and a good friend to me and my family.
I have been blessed with a wonderful group, the vvgroup. The group always gave
a feeling of being at home in the work environment. The group also actively contributed to the development of this work, through fruitful discussions and interactions
that helped me shape this work. I would also like to thank all my friends who helped
me in different ways at different times in completing this thesis.
This work was supported primarily by the MRSEC Program of National Science
Foundation under award number DMR 02-13282. I am also grateful to the department of materials science and engineering, for providing me with the excellent work
environment during the course of this work.
The prime motivation and support for my work, comes from my family, especially
my father and mother, who has been a source of inspiration for me through out my
life. My wife, Sheba has been the strength and support for me.
Contents
1 Introduction
13
2 Literature Review
17
2.1
Mechanical characterization of nano-scale films . ............
17
2.1.1
Effects of film thickness on estimated mechanical properties
18
2.1.2
Effects of indenter angle on mechanical properties .......
20
2.2
Effects of stresses exerted by focal adhesions on films of varying thickness 23
2.3
Mechanical characterization of living cells through modulus mapping
24
3 Modeling methods and geometry used for characterization of nanoscale films
3.1
27
Characterization of nano-scale films during nano-indentation ....
3.1.1
..
.......
.........
28
Modeling of indenter-substrate geometry to consider effects of
varying SPM indenter angle on measured mechanical properties
3.2
3.3
31
Modeling of substrate geometry to evaluate effects of focal adhesions
on substrate deformation ...................
......
33
Modeling of modulus maps . ..................
.....
34
4 Results and Discussion
4.1
27
Modeling of indenter-substrate contact for analyzing the substrate effects .......
3.1.2
.
37
Characterization of mechanical properties of thin films ........
4.1.1
Effects of film thickness on estimated E
. ...........
37
37
4.1.2
4.2
Effect of indenter angle on estimated E . ............
Effect of film stiffness and thickness due on focal adhesion induced
stress and strain fields
4.3
47
..........................
Modulus mapping of living cells ................
5 Conclusions and Future Work
53
..... .
58
63
List of Figures
2-1
Schematic of indentation setup in Hertzian formulation. ........
2-2
Schematic of cantilever tilted by an angle 0 with respect to the hori-
.
21
zontal surface...............................
2-3
20
Change in contact area as angle of indentation changes from 60 to 150
for an indentation depth of 25 nm and D = 25 nm. . ..........
22
3-1
Finite element model used in simulations of polymer film indentation.
28
3-2
Model showing 0Oangle of indentation. . .................
32
3-3
Bulk Model showing 60, 90 and 150 angle of indentation ........
3-4
Model of 50 nm thickness showing 60, 90 and 150 angle of indentation.
3-5
Finite element model of substrate used for simulation of cell-substrate
interactions ..................
.
..
..............
32
33
34
35
3-6
Image of the HUVEC cell. .......................
4-1
Finite element response of elastic film when t = 1000 nm and E = 0.5
MPa: (a) verification of Hertz model given in equation 3.1 and (b) von
M ises stress . . . . . . . . . . . . . . . . . . . .
. . .. . . . . . . . .
38
4-2 Von Mises stress distributions for films of varying thickness t and E =
10 MPa: (a) t = 1000 nm, h = 20 nm; (b) t = 600 nm, h = 20 nm;
(c) t -= 50 nm, h = 17.5 nm ...............
9
........
38
4-3 Comparison of experimental and calculated E for linear elastic films
as a function of film thickness t and constant indentation depth h =
20 nm: (a) E = 100 MPa; (b) E = 100, 10 and 0.5 MPa; (c) Deviation
from the Hertzian prediction of 3/2 power law for t = 50 nm and E =
100 MPa.
.................................
40
4-4 Comparison of E calculated using equation 3.1, assuming either a linear
elastic or a hyperelastic constitutive model (Mooney Rivlin, equation
3.3) for E = 0.5 MPa. ................
....
.......
41
4-5 Effect of indenter radius R on calculated E = 10 MPa: (a) as a function
of fraction of film thickness sampled f = hmax/t; (b) for a linear elastic
film of t = 50 nm indented to a depth of h = 12.5 nm with indenters
of nanoscale and microscale radii. . ..................
.
42
4-6 Demonstration of substrate proximity effects on overestimation of thin
film E, accentuated by the use of large indenter radii: (a) Calculated
E for an input modulus of 10 MPa and indentation depth of 12.5 nm;
(b) Corresponding load-depth (P - h) response compared to that of a
bulk sample of same input modulus; von Mises stress distribution for
film thickness t = 50 nm and h = 12.5 nm for (c) R = 2500 nm; and
(d) R = 25 nm .............
.
.............
43
4-7 Comparison of Hertzian and Dimitriadis [6] models of elastic modulus
calculation for input E = 100 MPa. Identical trends were observed for
E
4-8
=
10 MPa and 0.5 MPa .........................
44
von Mises stress distribution in bulk film for 0 = 0' and h = 13 nm. .
47
4-9 Von Mises stress distribution for 0 = 00 angle of indentation and h =
13 nm and t
=
50 nm thick film.
............
......
.
4-10 Effect of probe inclination angle 0 on estimation of E values. ......
48
48
.
4-11 Contact pressure distribution on a film of t = 50 nm and 0 = 15'
..
4-12 Contact pressure distribution as a function of t for different . .....
49
50
4-13 Shear strain distribution on a film of t = 50 nm and 0 = 150 for E =
0.5 MPa ..............
.
................
51
4-14 Shear strain distribution expressed as a function of S, on a bulk and
51
thin film for 60 angle of indentation . ..................
4-15 Shear strain distribution expressed as function of S for t = 50 nm and
0 := 60 , 90 and 150.
.....
...........
..
52
.........
4-16 Von Mises stress distributions for (a) four and (b) eight focal adhesions
54
per cell in a 0.5 pm thick film of E = 18 kPa. . .............
4-17 Distance to which the displacement field (u = 1 nm) is felt through the
thickness of the geometry as a function of E, for 4 focal adhesions/cell.
54
4-18 Distance to which the displacement field (u = 1 nm) is felt through
55
the thickness of the film as a function of E, for 8 focal adhesions/cell.
4-19 Magnitude of surface shear strain felt on top of the substrate as a
function of modulus for 4 focal adhesions/cell. . ...........
.
55
4-20 Magnitude of surface shear strain felt on top of the substrate as a
function of modulus for 8 focal adhesions/cell. . .............
56
4-21 Extent to which shear strains are felt when traction are exerted by
cells, for 4 focal adhesions/cell, expressed as a function of E. ......
56
4-22 Extent to which shear strains are felt when traction is exerted by cells
on the substrate, for 8 focal adhesions/cell expressed as a function of
.
elastic modulus E. ..................
.........
4-23 Displacement of cantilever vs deflection measured on the cell......
57
58
4-24 (a) Cell image showing the data points on which P - h curves were
taken. (b) Image of smaller section of cell used for analysis (mapping
modulus plot).. .........
...
................
59
4-25 Modulus values at various points on cell for the region shown in Figure
4-24(b).
........
..........................
60
Chapter 1
Introduction
Polymeric films of nanometer-scale thickness such as polyelectrolyte multilayers (PEMs)
and poly(dimethyl siloxane) (PDMS) are increasingly finding application in a wide
range of scientific fields ranging from bioengineering via synthetic cell substrata [14,
20] to tunable Bragg reflectors [9]. The mechanical properties such as (visco)elastic
moduli, play a very important role in defining the applications towards which these
nano-scale films can be applied. These properties are not expected to vary with length
scale but can be influenced through external artifacts of measurement. In the context
of biological applications, the mechanical properties of these films affect cell functions
ranging from mitosis to proliferation to programmed cell death [40, 13, 37]. The goal
of this thesis is to model the effects of finite thickness and contact conditions on the
accurate estimation of (visco)elastic stiffness for polymeric films including nanoscale
substrata and nanoscale cells. These contact conditions include pure normal loading
(indentation) and pure shear loading (cell focal adhesion traction).
One of the techniques that can test the mechanical properties of these materials
to nanoscale precision is scanning probe microscopy (SPM)enabled indentation. The
most common elastic contact formulations such as Hertzian [17] theory are used to
extract the mechanical properties of these materials, and are based on the assumption
that the material is continuous, homogeneous, elastic and relatively large in geometry
such that the external boundary conditions do not influence the estimation of a given
property. However, care must be taken in implementing these models to polymeric
films, which are usually non-linear elastic materials exhibiting time-dependent responses, such that external boundary conditions can significantly influence calculated
properties. One goal of this thesis is to decouple the geometric and mechanical properties of viscoelastic thin films, and show how each independently affects the stress
and strain fields within the film.
Because of the increasing application of SPMs to evaluate the mechanical properties via analysis of force-displacement responses, it is important to also evaluate
the experimental errors which could influence the calculated properties. One of these
errors is the inclination of the SPM cantilever with respect to the sample surface. The
tilt of the cantilever is necessary to ensure that the free end of the cantilever touches
the sample first. However this tilt influences the effective material stiffness(which is
a function of cantilever spring constant and tilt), inducing erroneous measurement
results [25]. Through systematic variation of cantilever angle and sample thickness,
the effect of varying each parameter is studied in detail.
In order to understand the geometric and material effects during mechanical testing or application of external loading situations, it is important to understand how
these stress and strain fields evolve.
One such example is the way in which cells
respond when the geometry and stiffness of the substrate to which they are attached
are varied[40]. The chemomechanical environment of living cells has been shown to be
influenced by these changes in external environment [24]. The influence of constant
stress which is exerted by a cell onto a polymeric substrate is evaluated as a function
of the polymer elastic modulus and thickness,as well as the number of cellular focal
adhesions (FAs).
Elastic modulus maps enabled by SPMs can be used to evaluate cell stiffness during various processes such as cell migration, or over different regions of the cell [13, 5].
Just as in the analysis of the film mechanical properties, the formulations from contact mechanics for deformable bodies are used in conjunction with force-displacement
responses collected from SPMs to estimate the elastic modulii at individual points on
the cell surface [13, 51, 21]. The concluding part of this thesis implements this tool
for evaluting cell stiffness over the cell surface.
Chapter 2 provides a brief literature review of mechanical modeling and characterization of polymeric films and cells. Chapter 3 details the computational modeling
pertinent to each study, and Chapter 4 presents Results and Discussion of these decoupled geometric and contact parameters for (bio)polymeric thin film mechanical
analysis. Chapter 5 enumerates the conclusions resulting from this study and discusses briefly possible ideas which originated during this thesis, that will be studied
as future work.
Chapter 2
Literature Review
Rapid advances in technology and process development techniques have made it possible to see, feel and apply nanometer scale objects in a wide variety applications in
science. This section discusses the advances made in characterization techniques for
mechanical properties of nanoscale polymer films and how these are applied to better
understand the internal characteristics of these films under various multiaxial loading
conditions. The progress made in the field of cell mechanics in understanding the
effects of loads (traction by cells) on these synthetic substrata are reviewed. Also reviewed are the research studies that combine cell mechanics with substrate properties
in creating the maps for local mechanical properties.
2.1
Mechanical characterization of nano-scale films
Mechanical properties such as elastic moduli are very important parameters that need
to be measured accurately in order to apply these polymeric films in various fields of
science. For thin films and polymers, estimated moduli are significantly affected by
external boundary conditions and experimental conditions. In the following sections,
the advances in understanding how these external parameters influence the calculated
modulii values are discussed.
2.1.1
Effects of film thickness on estimated mechanical properties
Mechanical properties such as elastic modulus E are measured from the load-deflection
(P - h) data extracted from nanoindentation. Studies in metallic and ceramic thin
films have shown that E is very strongly influenced by the thickness of the thin film,
and that calculated film properties differ very much from the bulk materials due to
this external artifact. The way in which contact-induced stresses are transferred to
the film/substrate interface and to the substrate itself very strongly influences the
estimated mechanical properties [32, 13]. The influence of the underlying substrate
on calculated mechanical properties has been addressed by several investigators for
conical indenter geometries [38, 11, 4, 46]. In this case, the contact stresses decay
sharply with distance from the indenter. This means that the mechanical properties
of the sample can be measured accurately if these contact stresses are totally confined
inside the film. Hence to capture only the response of the film, it is common to limit
the indentation depth hmax to less than 10% of the film thickness t. This is a purely
empirical estimate vary as a function of elastic and plastic mismatch between the film
and substrate [36]. These empirical relations may not hold when the sample property
exhibits non-linear elasticity or viscoelasticity.
The inherent geometric nonlinearity of multiaxial contact makes it difficult to obtain a closed form solution for contact of mechanically compliant samples that are of
finite thickness. The failure of Hertzian contact mechanics to estimate E from the
loading portion of the (P - h) indentation response for samples that are compliant
and of finite thickness has been addressed by others [6]. Dimitriadis et al. developed
a modified model which takes into account the effect of film thickness variation by
deriving the Green's function for a sample of finite thickness bonded to the substrate.
This model showed that for a given spherical indenter probe radius, there exists a
range of sample thickness t and indentation depth h.. The fundamental assumption
in these modeling techniques are that the material is a linear elastic solid, with the
maximum nominal contact strains not exceeding 10% or h < 0.1t. These models
are based on the early work of Chen et al., which addressed the problem of stresses
and strain fields in mulitlayer media that is well-adhered to the substrate [52, 44].
The Poisson's ratio v has been shown to influence E differently compared to models developed by Dimitriadis et al., when the finite thickness effect was considered
[21]. More conservative corrections for finite thickness were added through different
models, which also included the constraint of poorly adhered films. These models
estimated the complex elastic modulus and v of mammalian cells via SPM-enabled
nanoindentation. Although the modeling assumptions and approaches adopted for
developing these models differ, they all address the error in accurate determination
of E via contact-based measurements on thin, compliant polymer films.
For the specific case of polyacrylamide hydrogels of thickness t = 5 Am and for a
maximum indentation depth of hmax = 3 Am, it has been shown that the experimental
P - h responses agree well with the thickness-corrected Hertzian elastic cone model
proposed by Dimitriadis et al. In contrast, studies have shown that E for PEM of
jm-scale thickness is a function of t when calculated from SPM-enabled spherical
nanoindentation of hydrated, nanoscale PEM films even when this thickness correction is applied to the linear elastic Hertzian elastic sphere model [37]. Finally it has
also been demonstrated that E calculated from SPM-enabled spherical nanoindentation of hydrated, nanoscale PEM films with t < 200 nm, agrees well with the elastic
moduli calculated from other independent experimental techniques and also by other
researchers for similar PEMs [40] even in the absence of thickness correction factors.
An important question emerging out of these contrasting results is whether the E
of polymer films that are of nanoscale in thickness and measured via SPM-enabled
indentation are intrinsic properties of the material, or external artifacts due to combinations of finite thickness, large strains, and the inherent nonlinear elasticity of
the polymeric material.This becomes a critical issue when these polymeric films are
used for mechanics-critical applications that range from biological substrata to lowdielectric constant films in integrated circuits. Computational techniques such as
finite element analysis (FEA) can be used to analyze the mechanics of deformations
of these thin films which exhibit stress behaviors that are not confined inside the film
I4
L
Nanoscale PolymerFilm
-,,
"I
I
RiidSubstrate Figure 2-1: Schematic of indentation setup in Hertzian formulation.
geometry. The stress evolution beyond the elastic limit has been simulated via FEA
for thin films made of layered metal (titanium/aluminum) [22]. In this study, we simulate the nanoindentation of polymeric thin films using spherical indenters to analyze
the deformation field behavior which exists as a function of stress, film thickness, and
material constitutive model. Even though FEA are continuum based simulations that
are independent of length scale, this study provides important information about the
changes in stress and strain fields during nanoindentation when the thickness of these
films are reduced to the nanometer scale in thickness.
2.1.2
Effects of indenter angle on mechanical properties
The objective in section 2.1.1 was to understand how the geometric parameter such
as thickness of the sample influences E. This section discusses another critical question: Does the angle of indentation influence the estimation of mechanical properties,
especially for a case where thickness of the sample is on the order of nanometers?
Figure 2-1 shows the schematic for which the Hertzian formulation [17] is used. Here
the indentation is normal to the sample. However in conventional SPMs the indentation is imposed at an angle 0 as shown in figure 2-2. This tilt angle is usually around
120 to 150 in conventional SPMs. The Hertzian formulations described in the next
chapter do not account for this tilt during indentation.
Even though tilt of the cantilever is known to affect the properties being measured,
Figure 2-2: Schematic of cantilever tilted by an angle 0 with respect to the horizontal
surface.
most of the estimations of mechanical properties are based on the assumption that
the indenter is oriented normal to the sample geometry during indentation [15, 12].
In experiments such as studying adhesion of a particle (for example: when studied
in context of flow behavior of powders and granular media), the tilt of the cantilever
can have a significant influence on the absolute values of adhesion forces measured
[25, 27].Effective cantilever stiffness Keffective is increased by 10% - 20% as the tilt of
the cantilever increases [25], and is given by
Keffective
cos 2 0[1
-
K
2Dtan ]
(2.1)
where K is the stiffness of the cantilever, D is the diameter of the tip, L is the length
of the cantilever and 0 is the tilt of the cantilever with respect to the sample plane
[25].
When the cantilever is tilted, the force that is induced on the sample is not directed
perpendicular to the surface. The force components are split into one that is directed
parallel to the cantilever length and another directed perpendicular to the cantilever
length. The force that is directed parallel to the cantilever causes a torque, which
induces only a small change in deflection but a large change in the inclination of the
cantilever at, the end [25].
Figure 2-3: Change in contact area as angle of indentation changes from 6' to 15' for
an indentation depth of 25 nm and D = 25 nm.
One of the parameters which changes with indentation depth is the contact area
A, which is invariant with 0 for a sphere, but not for a sphere-tipped cone. The
calculations which are used to extract E are based on the assumption that the indenter
geometry is spherical in shape, which is not always the case. As the angle of inclination
changes, so does the contact geometry. A transition occurs from a spherical geometry
to a conical one. This fact is shown in figure 2-3, for an indentation depth of 25 nm.
It is seen that the geometry makes a transition from sphere (when 08= 60 )to conical
(when 0= 15)). This increase in contact area is an external artifact caused due to
the increase in tilt angle of the cantilever which affects the calculated mechanical
properties such as stiffness.The reason for this is explained below.
In most SPMs, the optical lever technique is used for detecting the cantilever
deflection. This means that the optical lever senses the cantilever deflection d and
not the inclination, from which the force is inferred as F = kd, hence making the
actual angle of inclination a critical parameter in property estimations. Correction
factors which are based on the geometry of the indenter have been proposed by several
researchers r25 , 19]. The correction factor is defined as the ratio of inclination of a
horizontal cantilever to the inclination of a tilted cantilever to which the same force is
applied. For a cantilever with a tip of conical shape this factor is cos20(1-2DtanO/L)
while the correction factor for a tip of spherical shape is cos 2 0(1-2RtanO/L), where R
and D are the indenter tip radius and the height of the cone respectively.
2.2
Effects of stresses exerted by focal adhesions
on films of varying thickness
The direct application of these thin, nanoscale polymer films is as synthetic substrata
for studying cellular function such as cell migration, cell adhesion, and cell differentiation. Among several topics, one area of interest is the interaction between cells and
substrates through focal adhesions, which are activated during transduction of local
mechanical forces that are rapidly converted into biological signals. Focal adhesions
(FAs) are large, mulitprotein complexes that provide a link between the cytoskeleton
and surrounding extracellular matrix [3]. The critical question we ask here is: How
does traction, exerted by the cells through FAs, affect the evolution of stress and
strain fields when the thickness and modulus of these films are varied?
Many researchers have studied cell functions over different types of substrates.
These substrates varied in their shape from patterned substrates to deformable substrates embedded with chemically functionalized micro-beads [35, 24, 52, 28].
By
using different synthesis techniques, E of these substrates are varied from the Pa
to MPa range [28, 40]. These studies have shown that the cells feel the underlying
substrate and actively respond to this change in local mechanical environment. The
rigidity of these substrates can regulate the formation and maintenance of tissues
[18], and the variation in mechanical properties of these substrates modulates the
mechanotransduction pathways [16].
The traction exerted by stationary fibroblasts has been shown to be a constant
2 [1]. Recently it has been demonstrated for fibroblasts that the
value of 5 nN/pmn
stresses exerted by these FAs increase exponentially when the area of FAs increases
beyond 8 prtm2 [30]. These increases in traction have also been associated with an increase in smooth muscle actin in fibroblasts, which is a sign of increased intracellular
tension. These mechanotransduction phenomena occurring inside FA depends on a
balance between the local mechanical stress and specific biochemical signals [29].
The sensing mechanisms differ from cell type to cell type. Fibroblasts show strong
adhesion and proliferation characteristics on substrates of very high elastic modulus,
while neurons grow best on more compliant substrates [50]. The fibroblasts showed no
stress fibers on softer substrates [50]. The cellular traction also increases with increase
in substrate E [30, 50]. Experiments have shown that the cells migrate preferentially
towards stiffer substrates from more compliant ones [35]. The mechanism through
which FA assemble, elongate and dissociate on application of external forces are not
well understood. It is shown that the force level, force distribution, and character of
anchoring of cells to the substrate all affect the FAs [47]. Studies on flexible silicone
substrates using beads as markers have shown that the forces exerted by motile fibroblasts onto their extracellular matrix environment are highly localized [49].
The demonstration that cells can deform the underlying substrate, and that substrate modulus can in turn affect cell behavior, indicates a coupled relationship between cells and the extracellular environment [41]. As the above studies show, there
is a great deal of experimental data available on the behavior of cells on deformable
substrates, but very little information is available on how the stress fields within
the substrates evolve when cellular traction is exerted. Understanding this evolution
would help to determine the range over which cells sample their local mechanical environment, and how thick substrata must be to dominate the mechanical environment.
This thesis addresses this question through computational modeling of the substrates
as thin polymeric film.
2.3
Mechanical characterization of living cells through
modulus mapping
SPMs can be used to measure the topography of materials and also the mechanical
properties such as elastic modulii and viscoelastic modulii. To understand cellular
physiology, it is important to know how mechanical properties vary with respect to
changes in time, space and various chemical processes. This can be analyzed by extracting E from the force deflection data measured from SPMs at many points on the
material (or cell) surface. Standard formulations developed for contact mechanics of
deformable substrates using rigid indenters [17, 51] are used to extract the E.
The viscoelastic characteristics of human platelets were measured using this technique [8] with E in the range of 1-50 kPa. The loss modulus which related to the
viscous response was two orders of magnitude lower than the storage modulus. The
cell relaxed during the load application and recovered fully when this external loading was removed, showing the recovery behavior of cells. The biomechanical changes
exhibited by synaptic vesicles were studied in detail using force maps. The modulus
values varied from 0.2 MPa to 1.3 MPa when these structures were exposed to various chemical environments. [34]. Hofmann et al. showed that the elastic moduli of
the lamellipodial region varied from 10 kPa to 200 kPa in living chicken cardiocytes
[33]. Force maps were used to measure the local mechanical properties of cytoskeletal
elements. The stress fibers were 3 to 10 times as stiff as the surrounding cytoskeleton.
The limitations of Hertz model in evaluating the force maps were also identified in this
study [33]. The extending and retracting dynamics of protruding and stable edges
of motile 3T3 fibroblasts showed that for the stable cell edges E was approximately
12 kPa. The leading edge had much lower modulus values equivalent to 3-5 kPa [7].
This study showed the limitations of using Hertz model to predict E, especially when
the thickness of the sample allowed substrate effects to dominate. These limitations
were also addressed in several other studies [8, 33].
The effect of various drugs on different cytoskeletal components of fibroblasts were
investigated using the elasticity maps [39]. The elasticity maps showed that the actin
filaments, which are the principal cytoskeletal component responsible for maintaining
cellular tension, disassociated and correlated with decreased cellular rigidity when
treated with certain drugs. These maps were used to study the modulus variations
amongst different clusters of cell surface macromolecules such as receptors and ion
channels [5]. The elasticity maps showed significant changes in local elasticity with
regions underlying the receptor clusters appearing less stiff. The modulus of outer
cell regions ranged from 50 to 120 kPa and around the nucleus ranged from 3 to 6
kPa. The E of human rhabdomyosarcoma transfectant cell line and NIH-3T3 mouse
fibroblasts showed sharp increases for even mild chemical fixation [19]. This study
showed how mechanical properties of cells changed during chemical fixation using the
modulus maps.
The above studies have shown that the modulus mapping technique is a very good
tool for understanding the variations in E during cell migration, changing mechanochemical environments and during cell fixation. This thesis implements this technique
of force volume mapping. The indention technique used to understand the stress fields
on thin polymer films in section 2.1 finds direct applications towards the way these
force-deflection curves are used for predicting the modulus values, as discussed in
section 3.3 of chapter 3.
Chapter 3
Modeling methods and geometry
used for characterization of
nano-scale films
This chapter discusses the modeling methods used in this study. Axisymmetric two
dimensional and three dimensional meshes are generated to evaluate the stress fields
evolving as a result of application of various boundary conditions such as for nanoindentation, tractive forces generated by cells etc. Also discussed in this section is the
approach used for modeling the modulus maps that are generated using SPMs forcedeflection data. The computational meshes are generated using meshing software
TRUEGRID from XYZ scientific applications [48] while the analysis is performed via
the general purpose finite element software ABAQUS [43].
3.1
Characterization of nano-scale films during nanoindentation
This section describes the modeling method used for analyzing the geometric effects
of thin films during indentation and also the effect of varying the angle of indentation
during these nanoscale indentations.
Figure 3-1: Finite element model used in simulations of polymer film indentation.
3.1.1
Modeling of indenter-substrate contact for analyzing
the substrate effects
Figure 3-1 shows the finite element model mesh and boundary conditions for the
simulations reported herein. This model included 34,476 nodes and 34,104 four-noded
axisymmetric elements.
Both the material films and the spherical indenter exhibit axial symmetry, and
are thus considered in a two-dimensional simulation. The indenter was modeled as
an infinitely rigid material of radius R = 25 nm, which is consistent with the nominal
apex radius of Si3 N4 pyramidal cantilevers used in SPM-enabled nanoindentation [40],
while the material films are modeled using linear elastic and, separately, hyperelastic
constitutive relations for film thicknesses of 50, 175, 300, 600, and 1000 nm. The distal conical portion of the cantilevered indenter is not modeled in these simulations,
as the typical maximum indentation depth implemented in related experiments [40]
and in these simulations is less than R [40]. The contact between the indenter and
the material film is assumed to be frictionless, and the nodes at the film/substrate
interfaces are modeled as rigidly fixed, based on the assumption that the polymeric
film is rigidly bonded to an infinitely rigid (or comparably so) substrate. A displacement controlled analysis is implemented to simulate the nanoindentation, and the
output extracted from the simulations is that of the loading force P corresponding to
the indentation depth h at each timepoint; quasistatic loading is thus assumed. The
maximum indentation depth hmax for all simulations is 20 nm, except for the model
for t = 50 nm where hax = 17.5 nm was the maximum depth at which the FEA stiffness matrix converged for the given mesh density, element type/shape function,and
constitutive relation employed.
The Hertzian elastic contact model is the most widely used closed-form solution
to calculate E of any material under generalized contact conditions. For small indentations (hmax << R), the paraboloidal indenter is well approximated as a sphere
for which the Hertzian elastic solution exists. The relation between the depth of a
spherical indenter and the corresponding applied load was formalized by Sneddon [42]
based on the Hertzian formulation:
4
Psphere
4-
E
E
.
(3.1)
where v is the Poisson's ratio of the sample (equal to 0.5 for ideally incompressible
materials,and assumed to be 0.49 for mathematical tractability in the simulations
herein). Thus E can be calculated directly from the output P-h response and chosen
simulated R for a given material constitutive relation via equation 3.1, although it is
important to note that equation 3.1 tacitly assumes linear elastic behavior of a semiinfinite, indented material. The nominal flow strain level e applied to the material
has been approximated in several forms, including: h/t, h/R and, most accurately, as
E = 0.2a/R.
(3.2)
where a is the radius of contact at the material surface [45]. The fraction of indentation depth h to film thickness t sampled is not an actual strain and is termed hereafter
as f, where the diameter of the elastically deformed contact zone and thus likelihood
from artifactual stiffening in P-h response due to the underlying substrate increases
with increasing f.
The polymeric films considered herein are modeled as both linear elastic and,
separately, hyperelastic materials. For the linear elastic case, three values of E were
considered: El = 100 MPa,E2 = 10 MPa and E3 = 0.5 MPa. These values are
based on the experimental results from Thompson et al., obtained via SPM-enabled
nanoindentation (Si3 N 4 cantilever of R = 25 nm) for thin film polyelectrolyte multilayers [40]. For the hyperelastic case, the strain energy function W for incompressible
materials was postulated by Mooney [31] and is given by
W = C1(I - 3)
+ C 2(/ 2 - 3).
where C, and C2 are the elastic constants and I1 and
(3.3)
12
are the first and second
invariants of the Cauchy-Green deformation tensor [2].
Based on the above formulation, for an incompressible material with infinitesimal
indentation depth h the Young's elastic modulus E can be expressed as
E = 6(Cl + C2).
(3.4)
Previous studies have shown that the widely applied infinitesimal strain models for
nanoindentation yielded substantial errors in the estimated properties such as E for
non-linear elastic materials [10]. The values for the elastic constants used in this
hyperelastic analysis are C1 = 0.0235 MPa and C2 = 0.060 MPa. These material
constants are not chosen to represent a particular polymer, but rather to assess how
the film thickness t and E independently affect the SPM-enabled nanoindentation
response for a constitutive model that is more representative of polymeric materials
than linear elastic models. The resulting value from equation 3.4 is E = 0.50 MPa,
which is consistent with one of the three linear elastic cases considered despite the
clear difference in the constitutive relations.
For a given film thickness t, there exists a critical depth of indentation h,, that
depends on the mechanical properties of both the film and substrate, beyond which
the force P required to attain a depth h increases due to the proximity of the film
/ substrate interface. This violates the semi-infinite film thickness (t >
hmax, R)
assumed by the Hertzian model, but h,, is not easily identified a priori [32, 26]. A
semi-analytical correction for this artifactual increase in stiffness due to finite film
thickness proposed by Dimitriadis et al. [6] as:
P = 16Ex /
9
[1 + 1.133X + 1.283X 2 + 0.769X3 + 0.0975x 4 ]
(3.5)
where
X
x=
h
(3.6)
is the correction factor due to finite thickness.The maximum film thickness for which
this correction is applicable is stated by Dimitriadis et al. as h < 12.8R [6]. This
restriction is based upon the assumption that the material behaves in a linear elastic
manner for maximum strains less than or equal to 10%,which may or may not be
true for a given material. Hence, in this study the thickness correction would apply
for film thickness t < 320 nm. Equation 3.5 can then be used to calculate E for
films of ostensibly identical or systematically varied mechanical properties, of varying
thickness. Mahaffy et al. [21] have shown that the value of the Poissons ratio also
affects the calculation of E when indenting samples of finite thickness. Experimental
results [21] have shown that as the Poissons ratio increases from 0.3 to 0.5, the depth
he, at which the underlying stiff substrate artifactually stiffens the measured response
actually decreases. This means that due to the high Poisson's ratio of polymers, even
in pim-scale thick areas of experimental samples such as living cells, the Hertz model
will fail to predict accurately the elastic properties.
3.1.2
Modeling of indenter-substrate geometry to consider
effects of varying SPM indenter angle on measured mechanical properties
This section deal with the modeling of inclined contact geometry for analysing the
effects of angle of indentation on the mechanical properties of the materials. The
indenter is modeled as a rigid material while the sample substrate is modeled as a
deformable material. The substrate is assumed to be incompressible. The model
is assumed to be linear elastic and isotropic. The angles of indentation 0 used are
00, 60, 90 and 150. For 0 = 00 angle of indentation only one quarter of the full model
is used due to the four-fold symmetry of boundary and loading conditions. Figure
3-2 shows the indenter and sample geometry at 0 = 00. For all the other cases where
angle of indentation is different from 00, half of the full geometric model is used. The
3-
Figure 3-2: Model showing 0Oangle of indentation.
Figure 3-3: Bulk Model showing 60, 90 and 15' angle of indentation.
reason for using the half-model is because both the boundary conditions and subsequently the stresses resulting from these boundary conditions are symmetric along
only one plane. The planes of symmetry here are x - y plane and y - z plane. Figure 3-3 shows the indenter and sample geometry at 0 = 6', 90 and 15'. In order to
understand the effect of thickness as the angle of indentation is changed, two cases
are analyzed. One case is where the thickness of the sample is equivalent to 300 nm,
which would be sufficient to model the material as bulk with the induced stresses
confined entirely within the geometry. The boundary conditions used for modeling
indentation are free slip between the indenter and the substrate during contact. For
evaluating the effect of thickness, a thin film of 50 nm thickness is modeled, as shown
in figure 3-4. For estimating E, the Hertzian formulation shown in Equation 3.1 is
used. The methodology as followed in section 3.1.1 for the axisymmetric case is used
Ir
~I
Figure 3-4: Model of 50 nm thickness showing 60, 9' and 150 angle of indentation.
here.
The two dimensional concept is extended to the three dimensional (3D) analysis.
Eight noded brick elements are used to model the geometry. In 3D, 24,471 nodes and
21,978 elements are used for modeling the bulk and 50nm thick samples during the 0'
angle of inclination 0. Here, 41,860 nodes and 38,016 elements are used to model the
bulk sample for 6', 90 and 15' angles of indentations while 48,300 nodes and 43,956
elements are used to model the 50 nm-thick sample for 8= 6' , 90 and 15' indentations.
3.2
Modeling of substrate geometry to evaluate effects of focal adhesions on substrate deformation
A three-dimensional finite element mesh made of brick elements is modeled to study
the effects of forces exerted by focal adhesions on substrates. It is assumed that the
cell is semicircular with a radius of 7.5 ym when resting on a substrate. The cells
adhere to the substrate through focal adhesions and the total area of the focal adhesion is assumed to be 1 pm 2 [1]. Because of the assumption that the cell-substrate
geometry is circular in shape, only one quarter of the geometry is modeled. The
model is shown in figure 3-5. The material model is assumed to be linear elastic,
isotropic and incompressible. The total area in all the focal adhesions is conserved.
This means that if there are 4 focal adhesions and the total modeled focal contact
area is equivalent to 1 pm 2 , then the total modeled contact area for 8 focal adhesions
is also equal to 1 Inm2
Figure 3-5: Finite element model of substrate used for simulation of cell-substrate
interactions.
Displacement boundary conditions are prescribed to all the points on the focal contact points.. These prescribed boundary conditions are given in such a way that the
traction evolving from these would equal to experimental conditions [1] of at = 5
nN/pm2 . The total number of nodes used here is 16,464 and the number of elements
is 14,355 for modeling 4 focal adhesions per cell; while 28,668 nodes and 25,344 elements are used for modeling 8 focal adhesions per cell. The modulus of films used in
this study are E= 18 kPa, 180 kPa and 1800 kPa and the thickness of the films t are
6 pm, 3 pm and 0.5 im.
3.3
Modeling of modulus maps
Elastic modulus values of each point on the cell surface are calculated by evaluated
the force deflection curves derived from force volume mapping. The cells used are
human umblical vascular endothelial cells (HUVECs).
In force-volume mode, the
SPM cantilever is scanned across the sample [51] and the force spectrum recorded
at each step. Force maps of 32 x 32 points are created across a 85 pm by 85 pm
cell-substrate area. Figure 3-6 shows the image of the cell, for which modulus at each
point in the cell is evaluated. Each point shown in figure 3-6 is a force map, which
Figure 3-6: Image of the HUVEC cell.
consists of 32 x 32 force curves. The scanning speed is 1.49 lines/sec which gives the
total time for measuring as 25 minutes. This time period is very large when living
cell imaging is done. Many cell types will change intrinsic mechanical characteristics
over this time period [51]. However, this factor is not considered here as this study
attempts only to implement modulus mapping as a general technique. The stiffness
,of the cantilever k is 0.06 N/m,and the half angle of the conical tip is 380.
To extract E from the force curves, Sneddon's and Hertzian model for elastic
indentation on a flat, soft sample by conical geometry is used [17, 42]. Based on
the cantilever deflection, and the formulations for contact mechanics developed by
Sneddon, the equation for conical tip can be expressed as follows,
h=
E 2 tan
(3.7)
On soft samples, the cantilever deflection d(z) is a function of piezo movement z and
indentation depth h given by
d(z) = z - h.
(3.8)
The force exerted by the cantilever P is given by
P = kh.
(3.9)
where k is the stiffness of the cantilever [51].
Combining equations 3.7,3.8,3.9 the final form used is
d- do = z- zo-
d- do.
(3.10)
where zo is the contact point at which the sample starts showing deflection and do is
the free cantilever deflection [19]. The value s is given by
8
r 1 -v
E
2
ta
k
8 E tana'
(3.11)
From equation 3.10 ,E is evaluated from each force curve through a non-linear fit [23].
Hence, by extracting E at each point, a complete modulus plot of the cell surface can
be obtained.
Chapter 4
Results and Discussion
4.1
Characterization of mechanical properties of
thin films
4.1.1
Effects of film thickness on estimated E
Results
The variation of load P with depth h on a log10 scale is shown in figure 4-la, for
film thickness t = 1000 nm and E = 0.5 MPa. The slope of 3/2 is consistent with
equation 3.1, which is the Hertzian formulation.
The value of E calculated is within 10% of the input value of E, which confirms
the well known result that for a linear elastic material of given E and film thickness
t, the contact response can be approximated by the Hertzian model. The von Mises
stress distributions shown in figure 4-1 b are well confined within the film.
The distributions of von Mises stress contours for a linear elastic case of E = 10
MPa and a range of film thickness t = 50, 600 and 1000 nm are shown in figure 4-2.
These simulations show how stress fields are transferred through the thin films, for
a given value of E. The stresses are confined to approximately to 10% of t. This
shows that for a given critical thickness, E can be predicted to within 15% of the
actual value, using the Hertzian formulation given by equation 3.1. The fundamental
4.6
4.8
5
5.2 5.4
5.5
log h [nm]
5.6
6
62
Figure 4-1: Finite element response of elastic film when t = 1000 nm and E = 0.5
MPa: (a) verification of Hertz model given in equation 3.1 and (b) von Mises stress.
[I·
Figure 4-2: Von Mises stress distributions for films of varying thickness t and E =
10 MPa: (a) t = 1000 nm, h = 20 nm; (b) t = 600 nm, h = 20 nm; (c) t = 50 nm, h
= 17.5 nm.
assumption while calculating E is that the material is linear elsatic. When the thickness is below the critical value, the underlying rigid substrate begins to influence the
prediction of E which is shown in figure 4-3. E calculated from the P - h response
using equation 3.1 is shown in figure 4-3 as a function of t. When the film thickness
is on the order of maximum depth of indentation hmax, this results in a significant
increase in calculated E with respect to the actual E that is dictated by the constitutive equation. Figure b shows the trend to be similar for all values of E (due to the
linear elastic assumption). Figure 4-3c shows the value of power law of equation 3.1
for a film that is of t = 50 nm and E = 100 MPa. The value of 1.67 deviates from the
theoretical value of 3/2, demonstrating the lack of applicability of the model. The
model which was originally intended for macro-scale contact mechanics, or for linear
elastic case with semi-infinite material dimensions, does not apply here due to the
film finite thickness.
Many polymeric materials are highly nonlinear in their behavior within the elastic
regime. In order to understand the non-linear material behavior, the Mooney-Rivlin
model has been applied. This shows that the overestimation of E is very similar in
behavior to the linear elastic case. These data demonstrate that the finite thickness
effects are seen in estimations of E, even when modeled using constitutive material
models which are more representative of polymers. Figure 4-4 that the calculated E
is close to the bulk or input value for t > t,, - 200 nm and maximum indentation
of hmax = 20 nm (Emax = 40%). Figure 4-4 shows the calculated E to be close to the
bulk value for a t > tcr = 200 nm with a maximum indentation depth of 20 nm. The
value E is overpredicted for both constitutive models, the linear elastic and hyperelastic cases. The tcr at which E is overpredicted increases for hyperelastic model.
That is, the nonlinear elastic deformation of the material relaxes the film thickness
constraint.
Figure 4-5 shows the variation of E with respect to the dimensionless parameter
f = hmax/t. For samples that are of nanoscale thickness (eg. t=50 nm) or when
maximum indentation depth hmax > 0.1t, the calculated E increases by more than a
factor of two for larger indenter radius R. Here, R = 2500 nm and 25 nm, for the same
(a)
200
180
160
160
14o
=•
a-
140
2
W
120
tu
I
12o
100
80
200
II
400
800
600
t[nml
1000
--r..............I
--F---r--r
N.
N.
I · · · · · · · · I··I··I·II·
0
200
400
· · II
600
t [nm]
800
1000
12
00
I
4.8
5
5.2
5.4
5.6
log h [nm]
5.8
6
6.2
Figure 4-3: Comparison of experimental and calculated E for linear elastic films as a
function of film thickness t and constant indentation depth h = 20 nm: (a) E = 100
MPa; (b) E = 100, 10 and 0.5 MPa; (c) Deviation from the Hertzian prediction of
3/2 power law for t = 50 nm and E = 100 MPa.
I
r-
FEM(LinearElastic)
0.8
FEM (Mooney
Rivlin)
0.8
08
•0.7
2 0.7
uJ
0.8
06
~I
0.5
n.
0
200
400
600
800
1000
1200
tInm)
Figure 4-4: Comparison of E calculated using equation 3.1, assuming either a linear
elastic or a, hyperelastic constitutive model (Mooney Rivlin, equation 3.3) for E =
0.5 MPa.
value of f. Naturally, strains induced by the larger indenter radius are much lower
compared to smaller indenter radii [6]. However, the calculated E are very high for
larger radii (figure 4-5b). The differences in calculated E values when a 50 nm thick
sample is indented to hmax = 12.5 nm indenter radii R of 25 nm and 2500 nm are
shown in figure 4-5b. Figure 4-6a shows the difference in estimated E when compared
to the actual value of E = 10 MPa. Figure 4-6b shows the P - h response for the
bulk and t :=50 nm samples when these are indented to h = 12.5 nm. Overprediction
of E occurs due to the substrate proximity effects which are shown in figure 4-6b-d.
When compared to the smaller indenter radii, the larger radii induce lower stresses
when indented to same depth. However, the transmission of stresses from the larger
indenter radii to the substrate is different because of the correspondingly larger contact zone of elastic deformation. This critical factor brings out a unique correlation
between the indenter geometry and shape of the stress contours for a given substrate
thickness. The stress distributions become flatter as the indenter geometry becomes
larger, spreading in the lateral direction compared to the loading axis as is evident
from figure 4-6d. One more factor that contributes to the overprediction of E is the
fact that the nanoscale films are very strongly adhered to the support structure. Due
25 nm
V.
0.1
0.2
0.3
0.4
I.
400
300
200
200
R [nm]
Figure 4-5: Effect of indenter radius R on calculated E = 10 MPa: (a) as a function
of fraction of film thickness sampled f = hmax/t; (b) for a linear elastic film of t = 50
nm indented to a depth of h = 12.5 nm with indenters of nanoscale and microscale
radii.
200
100
so50
1500oo
t [nm]
4
1.410
(b)
4
1.21o
,l.o0
-
Bulk
1 l0,
z
r
0,o'
610'
410'
I
h[nmi
15
2
5.00
2.50
O.00
MPg
Figure 4-6: Demonstration of substrate proximity effects on overestimation of thin
film E, accentuated by the use of large indenter radii: (a) Calculated E for an input
modulus of 10 MPa and indentation depth of 12.5 nm; (b) Corresponding load-depth
(P - h) response compared to that of a bulk sample of same input modulus; von
Mises stress distribution for film thickness t = 50 nm and h = 12.5 nm for (c) R =
2500 nm; and (d) R = 25 nm.
Hertz
180
-
Dimitriadis
S
Input
160
0.
140
120
100
I
an
0
1 i I 1 I I I I I I I I I I I I i i I I~ I I I I I .I
50
100
150
200
t [nm]
250
I I LUL
300
350
Figure 4-7: Comparison of Hertzian and Dimitriadis [6] models of elastic modulus
calculation for input E = 100 MPa. Identical trends were observed for E = 10 MPa
and 0.5 MPa.
to the smaller thickness along the loading axis, the material more strongly resists
lateral deformations during the indentation process, inducing larger stresses. This
becomes a critical factor when the indenter radius becomes large enough to increase
the contact stress zone both laterally and vertically.
Models proposed by Dimitriadis et al. [6] directly consider the effects of finite
thickness of mechanical properties estimated from the the P - h response. Figure
4-7 shows how the classical Hertzian model formulated using equation 3.1 compares
with the semi-analytical model for a linear elastic material which shows the variation of E with t. This model is superior to the Hertzian model in estimating the
E. However even this model overpredicts E below a certain value of film thickness.
This comparison between these analytical models is important for two reasons. First,
a model which was originally developed for macroscopic contact can be extended to
linear elastic samples of finite thickness if a finite thickness correction factor is used
[6]. Secondly, as shown in figure 4-7, even with thickness-corrected Hertzian formulations, overestimation of E as the thickness of t is reduced to the nanoscale. Table
4.1.1 shows the effect of dimensionless parameter f on the amount of error in calculated E when the film thickness is reduced or the depth of indentation is reduced, for
two indenter radii.
Effect of increase in f on contact strain % and predicted E expressed as a function
of indenter radius R.
t[nmn]
f
R[nm]
E
Eipet[MPa]
18000
0.03
2500
11.69
10
6
9000
0.07
2500
13.49
10
16
5250
0.11
2500
13.98
10
32
1500
0.40
2500
38.86
10
255
1000
0.02
25
28
10
10
600
0.03
25
28.29
10
12
300
0.07
25
28.26
10
16
175
0.11
25
30.25
10
24
50
0.35
25
43
10
89
Overestimation of
[%]
It is seen thlat both e as well as simulated E increase as f increases. As the value
of f becomes greater than 0.1, the error increases significantly, with % error in E
of approximately 255% for f = 0.4 for R = 2500 nm, and 89% for f = 0.35 when
R = 25 nm. The reason for this significant increase in error, especially for R =
2500 nmi, is because of the manner in which the stress fields redistributes around the
indentation zone as shown in figure 4-6c. The results discussed above are derived
based on several simplifying assumptions, including axisymmetric (rather than fully
three dimensional) geometry, and neglect of viscoelastic material models.
Discussion
The technological and processing advancements in materials processing of films have
enabled fabrication of films that are on the order of 100 nm or less in thickness. This
makes possible a wide range of applications such as synthetic substrata for understanding cellular functions [37]. This thesis addresses the applicability of Hertzian
analysis in estimation of E for polymeric thin films, when these films are deformed
through multi-axial loading conditions.
Figure 4-2 a-c shows that below a critical thickness ter and for a given depth of
indentation hmax, the Hertzian model is valid, provided linear elasticity is an accurate
material model. This means that the accurate estimation of E is possible, provided
that the depth of indentation is below h, for a given film thickness. The measured
value of E is accurate for indentation depth less than 20 nm which is the critical
thickness (for radius of indenter = 25 nm and E greater than 0.5 MPa) , beyond this
thickness the measured value of E is within 15% of the actual bulk value. The reasons
for this overestimation of E for larger indentation depths are enumerated below.
As shown in figure 4-3c the power law exponent derived from the Hertzian formulation starts to increase as the film thickness decreases from 1000 nm to 50 nm. This
deviation of hmax greater than 10% of film thickness actually increases the calculated
E values due to the proximity of substrate. This inaccuracy of E becomes a critical
issue when the substrate compliance contributes significantly to cellular morphology
and physiology [40, 35]. The trend of overestimated E is seen even for the case of a
hyperelastic material model, which ideally would resemble the polymer deformation
more closely than linear elastic model. As seen from figure 4-4, for E = 0.5 MPa. In
a hyper elastic material, tcr is reduced from 300 nm to 200 nm. This shows that E
can be extracted accurately for a larger value of f, without substrate artifacts, when
this model is used.
Figure 4-6c-d shows that the overestimation of E occurs even with larger indenter
radius. Larger radii induce smaller strains, but the stress field distributions are much
different for larger R as the film thickness decreases the transfer of stresses to the
substrate also increases. This demonstrates that even though the larger radius of indenters induces lower contact strains, this does not necessarily imply a more accurate
estimation of E. The overestimation of E with increasing R is shown in figure 4-5 a-b.
For accurate measurement of mechanical properties, it is ideal if the indenter radius is large enough to maintain mechanical strains below the elastic limit. This is
not often the case, as most cantilever probe radii are of the order of 25 nm. Such
small R induce strains that are greater than 10% for indentation depths that are less
than 20 nm. For such indenter geometries, the Hertzian model applies for t > 300
nm, as shown in figure 4-3a. Even for contact strains in excess of an estimated elastic
+0.1
+ 0.00 MPa
Figure 4-8: von Mises stress distribution in bulk film for 0 = 0' and h = 13 nm.
limit, the overestimation of E does not exceed 89%, as shown in Table 4.1.1. This
indicates that changes greater than this could be due to real microstructural changes
or external artifacts other than film thickness t.
4.1.2
Effect of indenter angle on estimated E
Results
While section 4.1.1 focused on understanding the effect of thickness on estimations
of modulus, this section will enumerate the results from the analysis for indentation
at inclination angles 0 = 00, 60, 90 and 150. The maximum indentation depth hmax in
all cases is 13 nm. The indenter and sample geometry for bulk and thin film cases of
various inclination angles are shown in figure 3-2, figure 3-3 and figure 3-4. As shown
in figure 2-3, as the angle of indentation increases the contact area between the
indenter geometry and film also increases. This increase in contact area subsequently
affects the accuracy of calculated mechanical properties [25]. This section decouples
the effect of angle of indentation during nanoindentation from finite film thickness.
Figure 4-8 shows the von Mises stress distribution for 0 = 00 for a thick(pim-scale)
film. This shows that the stresses are confined inside the film. Figure 4-9 shows the
von Mises stress distributions for 9 = 00 for a film of t = 50 nm. This figure clearly
Figure 4-9: Von Mises stress distribution for 0 = 0Oangle of indentation and h = 13
nm and t = 50 nm thick film.
0.90
0.80 _
0.70
0.60
A 0.50
*Bulk
*
50 nm
0.40
0.30
0.20
0.10
0.00
0
5
10
15
20
0 (deg)
Figure 4-10: Effect of probe inclination angle 0 on estimation of E values.
shows the stresses to be no longer confined inside the film. The modulus value E
does not change with 0 as shown in figure 4-10. All the calculations for E are based
on the assumption that the indenter geometry is spherical. During experiments, this
assumption does not hold true as seen in figure 4-10. This clearly demonstrates the
deviation from spherical geometry during indentation at larger 0. The invariance of
calculated E as a function of 0 holds true for both the bulk geometry and for the thin
film case where t = 50 nm. As seen from figure 4-10 for the bulk case, the calculated
value does not change from the input or actual E value by more than 10%.
However, for the thin film case (t = 50 nm), there is an increase in estimation of E
Figure 4--11: Contact pressure distribution on a film of t = 50 nm and 0 = 150.
as shown in figure 3-3. This increase in E when compared to the bulk value is because
of the decrease in film thickness and relative increase in substrate proximity, leading
to overestimation of E. However the value of E does not change with 0 even for the
thin film case, which demonstrates that 9 does not greatly affect the estimation of E
and it is only the change in film thickness that strongly affects the calculated E.
Figure 4-11 shows the von Mises stress distribution for the case of t = 50 nm and 0
= 15'. As shown in Figure 4-12, the value of peak contact pressure Pmax remains the
same for a given film thickness. The force range during typical scanning process using
SPM is of the order of 1 nN. This force range has been found to induce destructive
effects and changes in cellular morphology eventually leading to cell death [8, 53]. The
contact pressure distribution shown in figure 4-12 corresponds to an average force of
100 pN for hmax = 13 nm.Here, Pmax does not change with the 0, but does increase
with decreasing t. It is seen from figure 4-12 that for a given t, p,,ax is constant with
0.
One of the critical parameters of interest is the manner in which shear strains
change as a result of changes in 0 and t. The behavior of changes in these strain fields
0.45
0.4
•
-
0.35
E. 0.3
i"'
.
SBulk
50 nm
0.25
0.2
1 0.1
0.1
0.05
0
T
0
2
4
6
8
10
12
14
16
0 (deg)
Figure 4-12: Contact pressure distribution as a function of t for different 0.
are important because of several reasons. First the distribution of these surface strains
would help to understand how indentation affects the regions near the indentation
area which would subsequently help in designing polymeric films that can limit or
increase the strain fields due to applied loads. Secondly, the range of E analyzed in
this thesis is similar to that of substrates that serve as extracellular matrices for cells.
Hence, thickness-dependent changes in strain field distributions of these thin films
may help to better understand how strain is transmitted under cellular traction.
Figure 4-13 shows the shear strain distributions (612) for the case of a thin film.
The strain field extends down to the bottom of the film with both positive and
negative shear strains acting through the film. This figure also shows that during
nanoindenation, the multiaxial loading state causes the the maximum shear strain
to exist not on the surface but within the film. Figure 4-14 shows the magnitude
of
F12
for 0 = 60, expressed as function of film thickness along the indenter profile
in direction normal to the film surface.
The figure shows that the magnitude of
shear strain increases as thickness t decreases. The width over which shear strain
is non zero, is defined as distance S in figure 4-13. S is - 50 nm for the bulk film
case. As the film thickness decreases, S increases to 80 nm. This shows that both
the magnitude and extent to which shear strain extends increase with decreasing t.
Figure 4-15 shows the magnitude and spread of shear strains S as a function of 0.
Figure 4-13: Shear strain distribution on a film of t = 50 nm and 0 = 150 for E =
0.5 MPa.
OS
04
0.3
0.2
j/
t -0
-bul=5krl
-02
-0.
-04
05,
SfnnJ
Figure 4-14: Shear strain distribution expressed as a function of S, on a bulk and
thin film for 60 angle of indentation .
0.5
0.4
0.3
---.
B= 6°
0.2
0.1
-
= 15.
0
00012 0.00014
-0.2
-0.3
-0.4
-0.5
Figure 4-15: Shear strain distribution expressed as function of S for t = 50 nm and
0 = 60,90 and 150.
Figure 4-15 demonstrates that neither the magnitude nor distribution of
c12
change
when 0 is changed. This means that for a given t, the magnitude and distribution of
612
are not affected strongly by 8.
Discussion
The parameters E, pmax, and 612 show no significant change as a function of 0. There
are several reasons for the this behavior, as discussed below.
Throughout this analysis, the maximum indentation depth was 13 nm, approximately equivalent to the indenter radius R = 25 nm. This means that even with
the change in 9 as high as 15', the spherical geometry would dominate the stress
field (and the conical part of the indenter would not contribute).
Because of this
equivalence, the contact geometry was invariant with 8 in all cases analyzed.
In this analysis, the indentation load application is allowed to occur normal to the
film whereas in experiments the load is applied at an inclined angle 0. This means
that there would be two components of force, one along the long axis of the cantilever
and one normal to the long axis of the cantilever. The component along the axis of
the cantilever is responsible for the drag and external torque during nanoindentation
[25].
The contact pressure distribution and maximum contact force exerted are critical
parameters during nanoindentation. During the nanoindentation process, for cells to
exhibit recoverable deformation behavior it is essential to keep the indentation forces
below 1 nN [53]. It is seen here that the forces are below this level during the nanoindenation process, hence demonstrating the fact that during nanoindentation, if the
indentation depth is kept to a point where only a spherical geometry dominates the
indentation contact, cells could be indented non-destructively.
4.2
Effect of film stiffness and thickness due on
focal adhesion induced stress and strain fields
Results
As described in Chapter 1, one direct application of these nanoscale polymeric films
is in the field of bioengineering. These films are used as synthetic substrata to culture
cells and subsequently understand cellular physiology as a result of variation in the
mechanical properties of these substrates. These films will be under multiaxial loading conditions when adhered cells are fully confluent, due to the traction exerted by
the cells. The aim of section 4.1.1 and 4.1.2 was to quantify the effect of film thickness
and angle of indentation on calculated film properties. In this section, we attempt to
characterize the deformation of these films, now cell substrates, as a function of focal
adhesions as a function of film E.
The range of E and the type of boundary conditions used in this study are described in chapter 3. For this study the focal adhesion traction stresses were in the
range of 3 nN/pm 2 - 5 nN/,lm
2
[1]. Figure 4-16 shows the von Mises stress distri-
butions for the case of t = 0.5 im , when the prescribed displacements are applied
to obtain the desired traction. The boundary conditions prescribed are on a film of
elastic modulus E = 18 kPa. As seen from figure 4-16 the stresses exerted on these
films are highly localized. This agree well with experimental studies that showed that
the stresses exerted by cells onto the substrate through focal adhesions are highly
localized and do not spread over a large area [1, 29]. One of the critical questions this
20
S10
0.00KPa
Figure 4-16: Von Mises stress distributions for (a) four and (b) eight focal adhesions
per cell in a 0.5 pm thick film of E = 18 kPa.
4I
3.5
3
-
2
---
1.5
t =m6gm
t = 3 -m
-t =0.5
pcm
0.5
1.00
1.50
2.00
2.50
3.00
3.50
[kpa])
Log (E
Figure 4-17: Distance to which the displacement field (u = 1 nm) is felt through the
thickness of the geometry as a function of E, for 4 focal adhesions/cell.
thesis attempts to answer is how the thickness of the film influences the stress and
strain fields within the film due to cellular focal adhesion-induced surface traction.
Figure 4-17 and figure 4-18 show the displacement field through the film thickness,
when similar tractions are exerted by 4 or by 8 focal adhesions per cell. These figures
show that the extent to which the film displaces decreases as E increases, regardless of
the number of FAs. The data collected through the thickness are at the points directly
normal to t he focal adhesion area, where the displacement field is maximum. Figure
4-17 and figure 4-18 suggest that the change in the extent to which the displacement
field extends is not significantly different between 4 and 8 focal adhesions/cell, and
the trend followed by both as a function of E are similar. However, for the case of
films with thickness of t = 0.5 pm, the displacement field extends through the entire
-5
r4
-~~-t=3p.5m
3=
3
0*
1.00
1.50
2.00
2.50
3.00
3.50
Log (E (kpal)
Figure 4-18: Distance to which the displacement field (u = 1 nm) is felt through the
thickness of the film as a function of E, for 8 focal adhesions/cell.
7
m
5
m
4
mm:
3
2
1.00
1.50
2.00
2.50
3.00
3.50
log (EIkpa])
Figure 4-19: Magnitude of surface shear strain felt on top of the substrate as a
function of modulus for 4 focal adhesions/cell.
film thickness. This means that there will be finite thickness effects for films of E <
18 kPa for t < 0.5 pm.
One of the parameters of interest in tissue engineering is the magnitude and distribution of strains due to traction exerted by living cells against a given substrate.
Figure 4-19 and figure 4-20 quantify the shear strain on the surface as a function of
distance from focal adhesions.
The trend of shear surface strains exerted by focal
adhesions is the same for both the four and eight focal adhesions/cell, with the magnitude of shear strains decreasing as E increases. It is seen that in the case of 8 focal
adhesions/cell, the strain magnitude is almost an order of magnitude greater than
that for 4 focal adhesions/cell, even though the total applied traction in both these
cases was similar.
45
40
35
30
S25
20
15
10
5
0
0.00
nm
nm
imm
0.50
1.00
1.50
2.00
log (E(kpal)
2.50
3.00
3.50
Figure 4-20: Magnitude of surface shear strain felt on top of the substrate as a
function of modulus for 8 focal adhesions/cell.
3.5
2.5
C 2
"*'ý
ý11,
1.5
0
t=6pm
i
t 0.5I
5w
m
0.5
0
0.00
1.00
2.00
3.00
4.00
log(E[kpaD
Figure 4-21: Extent to which shear strains are felt when traction are exerted by cells,
for 4 focal adhesions/cell, expressed as a function of E.
The extent to which surface shear strains extend is quantified in figure 4-21 and
figure 4-22.
Both figures show similar trends, regardless of the number of FAs/cell.
Also, the extent to which the shear strains extend in 4 FAs/cell are greater than 8
FAs/cell for a given E and t. This could indicate that the spread of the strain field
would decrease as the number of focal adhesions increases, and the force is redistributed in the cell.
2.5
S1.5
i
--.
5---t
t=6 Pm
= 6 pm
0.5
0.00 0.50
1.00
1.50 2.00
2.50
3.00
3.50
log(E[kpa])
Figure 4-22: Extent to which shear strains are felt when traction is exerted by cells
on the substrate, for 8 focal adhesions/cell expressed as a function of elastic modulus
E.
Discussion
Figure 4-17 and figure 4-18 show the extent to which the displacement extends through
the substrate. This means that as the number of focal adhesions increases (typically
around 300 for fibroblasts [30]), the extent to which the focal adhesion-induced displacement is affected by the underlying film thickness varies. The combined effect of
all these focal adhesions from a single cell onto the film remains to be analyzed.
The magnitude and extent of surface shear strains exerted are based on few focal
adhesions in this study. In actuality there are millions of cells each having hundreds
of focal adhesion points anchoring to the film. This may induce film deformation that
might differ from what is proposed here. Even then, this study provides preliminary
results on how film thickness and elastic stiffness affect transmission of forces due to
cell adhesion. The FA area may also play a major role in exerted stresses and strains.
This thesis develops all results based on a single combined area of 1 Pm 2 exerting at
= 5 nN/plrr 2 . In actuality, this value differs from FA to FA and also varies among
different cell types.
The reason for decoupling E and t here to study cellular adhe-
sion is because, in ideal experimental conditions, there are several variables that need
to be considered which would make analysis extremely complex and time consuming
such that it would be difficult to infer geometric and mechanical effects from a cell
population study.
1100
900
700
.
500
300
100
-100
0
1000
2000
3000
4000
5000
6000
7000
Displacement of the cantilever [nm]
Figure 4-23: Displacement of cantilever vs deflection measured on the cell.
4.3
Modulus mapping of living cells
Force-volume mapping is one of several tools which can be used to characterize E
of thin films and of cells. The topic of interest in previous sections was to extract
and evaluate the mechanical properties of polymeric thin films, using formulations
based on contact mechanics when these films were subjected to multiaxial loading
conditions. This section deals with extracting E from the P - h responses obtained
from force-volume experiments.
Figure 3-6 shows the region on a human umblical vascular endothelial cell (HUVEC),
on which the P - h responses were obtained. Figure 4-23 shows the deflection versus
displacement of cantilever for one point on the cell. Equation 3.10 (conical indenter,
discussed in Chapter 3) is used to extract E at each such point on the cell. A small
section of the cell shown in figure 4-24 is used for demonstrating the extraction of E.
The curves as shown in figure 4-23 are normalized such that the height and deflection
values start from 0. The slope of the curve changes as soon as the cantilevered probe
comes in contact with the cell. This point at which the value of deflection starts
Figure 4-24: (a) Cell image showing the data points on which P - h curves were taken.
(b) Image of smaller section of cell used for analysis (mapping modulus plot).
E [kPa]
70
65
7
60
55
E
6
I 150
45
0
40
0.
35
4
30
25
3
20
L
2
3
4
5
6
7
8
y position [plm]
Figure 4-25: Modulus values at various points on cell for the region shown in Figure
4-24(b).
The fitting of the curve starts from this point. The
increasing is labeled as z o.
0
Poisson's ratio is assumed to be 0.5. Even though this value will differ based on the
point of indentation, it is impossible to extract the Poisson's ratio independent of E.
The modulus of the points were in the kPa range, which is close to the values reported
in the literature [51]. Figure 4-25 shows E for the points shown in figure 4-24b.
Discussion
Modulus mapping of deformable polymeric films is a very good technique to extract
local mechanical properties. However, these properties are largely dependent on the
type of indenter geometry used for indentation, as well as the depth to which the
indent ation is made. For example, the thinner regions of the cell give a modulus
value of E higher than the actual value because of the finite thickness effects which
were discussed in detail in section 4.1.1. To accommodate this problem, lower forces
should be used for indentation [12]. The overprediction of E for polymeric films, due
to the choice of larger range of forces are shown in detail by Domke et al [12].
Another problem specifically encountered in this study was the lower number of
points in the data range chosen. This makes it difficult to fit the curve accurately.
This could be solved by increasing the number of points for the entire force-deflection
map. Regardless of these problems, the force-volume technique provides a means
and method to directly measure the elastic modulus of cells, which can subsequently
be used as a powerful tool for analyzing living cell moduli as a result of changes in
external environment.
Chapter 5
Conclusions and Future Work
The systematic computational evaluation of contact-enabled mechanical characterization of nano-scale thin film brings out several key features. In order to determine
E of these thin films, the effect of finite thickness and non-linear elastic deformation
should be included. The inclusion of these effects enables classical formulations derived for deformable contact bodies to be used to successfully and efficiently to extract
the E. Even when the strain fields exceed the nominal elastic values, the magnitude
of E does not exceed by an order of two. Also, the choice of larger indenter radii R
should be made with caution, because the manner in which stress fields evolve inside
the region of indenter-substrate contact during indentation changes significantly with
R, hence influencing the calculated E.
There is not significant difference in estimations of E when the angle of indentation 0 is varied from 0Oto 150 for hmax ; R. The effects that dominate during the
indentation on nanoscale films are that of thickness rather than angle of indentation.
This means that the factor that dominates the thin film indentation are the geometric
effects.
While using these thin films are synthetic substrates for analyzing cellular
functions it is seen that the effects of displacement field felt towards normal to the
focal adhesion are more dominant compared to the component along the substrate.
The effects of modulus variations are pronounced normal to the cellular traction and
towards the thickness of the substrate in comparison to the surface. This might mean
that the cellular strains exerted during cellular functions are felt more on the extra-
cellular matrix or on cells that are directly below the point of exertion of tractive
forces compared to cells that are adhered directly to the substrate.
The modulus mapping could be developed as a useful tool to analyze cellular and
substrate mechanical properties. These in turn would provide crucial clues to the
working of cell when its mechanical and chemical enviroment are varied.
The future work will be aimed at identifying how the bidirectional relations between cell and its local microenvironment vary at points of contact between cell and
the substrate. This would involve methods such as modifying the stiffness of the
underlying substrate, varying the chemical environment surrounding the cells and
looking at how these changes affect cellular mechanotransductions, using computational methods.
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